Publicity Campaign

Contents

Problem description

The small company, Pronuevo, launches a new product into a regional market and wishes to have a publicity campaign using different media. It therefore contacts a regional publicity agency, PRCo, that specializes in this type of regional campaign and completely hands over this task for a total budget of $ 250,000. The agency knows the market well, that is, the impact of publicity in a local magazine or over the radio, or as a TV spot on the regional channel. It suggests addressing the market for two months through six different media. For each medium, it knows the cost and the number of people on which this medium has an impact. An index of the quality of perception of the campaign is also known for every medium.

The publicity agency establishes a maximum number of uses of every medium (for instance, not more than eight transmissions of a TV spot). The table below lists all this information. Pronuevo wants the impact of the publicity campaign to reach at least 100,000 people. Which media should be chosen and in which proportions to obtain a maximum index of perception quality?

Data for the publicity campaign

+------+------------------+----------+------+-------------+----------+
|      |                  |People pot| Unit |Maximum      |Perception|
|Number|Media type        | reached  | cost |  use        | quality  |
+------+------------------+----------+------+-------------+----------+
|   1  |Free weekly newsp | 12,000   | 1,500| 4 weeks     |    3     |
|   2  |Monthly magazine  |  1,500   | 8,000| 2 months    |    7     |
|   3  |Weekly magazine   |  2,000   |12,000| 8 weeks     |    8     |
|   4  |Radio spot        |  6,000   | 9,000|60 broadcasts|    2     |
|   5  |Billboard 4x3 m   |  3,000   |24,000| 4 boards    |    6     |
|   6  |TV spot           |  9,000   |51,000| 8 broadcasts|    9     |
+------+------------------+----------+------+-------------+----------+

Variables

budget                     Budget
people                     Persons potentially reached per media
costs                      Unit costs
maxuse                     Maximum use
quality                    Perception quality
minpeople

Reference

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005.   Last modified Apr 8, 2009.

Problem setup

budget    = 250000;
people    = [12000 1500  2000 6000  3000  9000]';
costs     = [1500  8000 12000 9000 24000 51000]';
maxuse    = [4 2 8 60 4 8]';
quality   = [3 7 8 2 6 9]';
minpeople = 100000;

m = length(maxuse);
use = tom('use',m,1,'int');

% All variables are integer.
bnds = {0 <= use <= maxuse};

% Budget constraints
con1 = {costs'*use <= budget};

% Reach constraint
con2 = {people'*use >= minpeople};

% Objective
objective = -quality'*use;

constraints = {bnds, con1, con2};
options = struct;
options.solver = 'cplex';
options.name   = 'Publicity Campaign';
sol = ezsolve(objective,constraints,[],options);

PriLev = 1;
if PriLev > 0
    invest = sol.use;
    for   i = 1:length(invest),
        if invest(i) ~= 0,
            disp(['invest in ' num2str(invest(i)) ' uses of media number ' num2str(i)])
        end
    end
end

% MODIFICATION LOG
%
% 051201 med   Created
% 060116 per   Added documentation
% 060125 per   Moved disp to end
% 090308 med   Converted to tomSym

Problem type appears to be: mip
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Publicity Campaign             f_k    -122.000000000000000000
                                              f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv    1 
invest in 4 uses of media number 1
invest in 2 uses of media number 2
invest in 8 uses of media number 3
invest in 4 uses of media number 4
invest in 4 uses of media number 5